# On the motion of the free boundary of a self-gravitating incompressible fluid

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Shuang Miao , University of Michigan
Fine Hall 801

Please note special time and location of this talk.  The motion of the free boundary of an incompressible fluid body subject to its self gravitational force can be described by a free boundary problem of the Euler-Poisson system. This problem differs from the water wave problem in that the constant gravity in water waves is replaced by a nonlinear self-gravity. In this talk, we present some recent results on the well-posedness of this problem and give a lower bound on the lifespan of smooth solutions. In particular, we show that the Taylor sign condition always holds leading to local well-posedness, and for smooth data of size $\epsilon$ a unique smooth solution exists for time greater than or equal to $O(1/{\epsilon}^2)$. This is achieved by constructing an appropriate quantity and a coordinate transformation such that the new quantity in the new coordinate system satisfies an equation without quadratic nonlinearities.  This is joint work with L. Bieri, S. Shahshahani and S. Wu.